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  1. 003A portfolio holds 60% in a stock with a beta of 1.2 and 40% in cash. The index falls 10%. What move do you expect in the portfolio from market exposure alone?Correlation and diversificationWarm upAsset manager risk

    Try it first

    What is the expected move in the portfolio?

    Show the worked solution

    About 7.2% down. Portfolio beta is the weighted average of the holdings' betas: 0.6 times 1.2 for the stock plus 0.4 times zero for the cash, which is 0.72. Multiply by the index move of minus 10% and the expected move is minus 7.2%. The stock itself is expected to fall 12%, but the cash dilutes it.

    Why does the cash count as zero?

    Imagine a household where one earner's pay swings with the economy and the other's savings sit in a bank account. When the economy dips, only the first half of the income moves. Beta measures how much a holding moves for each 1% move in the index, and cash does not move with the index at all, so its beta is zero. The portfolio beta is then the weighted sum: 0.6 times 1.2, which is 0.72, plus 0.4 times zero.

    Portfolio beta is a weighted average, and cash counts as zeroWeights across Rs 100Stock 60%, beta 1.2Cash 40%, beta 0Contribution to beta = weight x betaStock0.6 x 1.2 = 0.72Cash0.4 x 0 = 0Portfolio beta0.72If the index falls 10%-10.0%Index-12.0%Stock-7.2%PortfolioBeta is the expected move per 1% move in the index, from market exposure only. Stock-specific news comes on top.
    The stock is 60% of the portfolio with a beta of 1.2 and contributes 0.72; the cash contributes nothing, so a 10% fall in the index maps to an expected 12% fall in the stock but only 7.2% in the portfolio.
    The relationship
    βp=∑iwiβi=0.6×1.2+0.4×0=0.72Δp=0.72×(−10%)=−7.2%\beta_p = \sum_i w_i \beta_i = 0.6 \times 1.2 + 0.4 \times 0 = 0.72 \qquad \Delta_p = 0.72 \times (-10\%) = -7.2\%
    w_ieach holding's share of the portfolio
    beta_ieach holding's sensitivity to the index
    What it says in wordsWeight each holding's beta by its share of the money, add them up, and scale the index move by the result.

    What does beta leave out?

    Everything that is not the market. Beta gives the expected move from market exposure; the stock's own news adds a separate, unpredictable move on top. On Rs 10 lakh the expected loss is Rs 0.72 lakh, about Rs 72,000, but the actual loss could be larger or smaller depending on what happens to that one company. A single stock position carries a lot of this idiosyncratic riskRisk specific to one company, such as a product failure or a management change, which does not move with the market as a whole., which is why a risk manager quotes beta as an expectation, not a forecast.

    Also say that beta is estimated from past returns and drifts over time. A stock measured at 1.2 over the last three years can behave like 1.5 in a sell-off, because correlations tend to rise when markets fall. The 7.2% is the right answer to the question as posed; the conversation that follows is about how much to trust the 1.2.

    Where candidates lose it

    Candidates answer 12%, which is the stock's expected move, and forget that 40% of the money is in cash. The question is about the portfolio, and the weights are the point.

    The second miss is presenting 7.2% as what will happen. Call it the expected move from market exposure, and name the stock-specific risk that sits on top.

    What the interviewer asks next

    • How much of the stock would you sell to bring the portfolio beta to 0.5?
    • The cash is replaced with a bond fund with a beta of 0.1. What is the new portfolio beta?
    • How would you hedge the market exposure with index futures, and what risk would remain?